This paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data. With appropriately graded temporal grids, first-order temporal accuracy in the norm of $L^2(0,T;L^2\left(\Omega \right))$ is derived by a new discrete Grönwall’s inequality, and then first-order temporal accuracy in the norm of $L^{\infty }(0,T;L^2\left(\Omega \right))$ is established for $1∕2<\alpha <1$. Finally, numerical results are provided to verify the theoretical results.

中文翻译：

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半线性分数扩散方程的时间离散化分析

本文考虑具有不光滑初始数据的半线性分数扩散方程的时间离散化。使用适当分级的时间网格，范数范数的一阶时间精度${\mathrm{大号}}^2（0，Ť;{\mathrm{大号}}^2（\Omega ））$ 由一个新的离散Grönwall不等式推导得出，然后由 ${\mathrm{大号}}^{\infty }（0，Ť;{\mathrm{大号}}^2（\Omega ））$ 为 $\mathrm{1个}∕2<\alpha <\mathrm{1个}$。最后，提供数值结果以验证理论结果。